Geordie Drummond McBain

Natural convection in horizontally heated ellipsoidal cavities is considered in the low Grashof number limit, solving the Laplace equation for steady thermal conduction in the unbounded solid exterior, the Oberbeck–Boussinesq equations in the fluid-filled interior, and matching the temperature at the interface. In the hierarchy of equations governi...

The immediate impulse-response of a confined incompressible fluid is characterized by inertance. For a vessel with inlet and outlet, this is a single quantity; for multiple ports the generalization is a singular reciprocal inertance matrix, acting on the port-impulses to give the corresponding inflows. The coefficients are defined by the boundary-f...

A simple lumped hydraulic model of knee drainage following arthroplasty is developed incorporating a pressure-volume equation of state for the knee capsule and a wound healing rate dynamically retarded by the blood flow-induced shear stress. The resulting second-order nonlinear ordinary differential system is examined numerically and qualitatively...

The stability of the buoyancy layer on a uniformly heated vertical wall in a stratified fluid is investigated using both semi-analytical and direct numerical methods. As in the related problem in which the excess temperature of the wall is specified, the basic laminar flow is steady and one-dimensional. Here flows varying in time and with height ar...

To investigate the effectiveness of the dispersed-phase continuum (DPC) approximation to model the airflow in the print gap of inkjet printers, three-dimensional simulations using the DPC model were compared against those using the classic particle-in-cell (P-in-C) approach. The DPC approximation, due to the separation of time scales, models the di...

The mass of individual droplets ejected from a thermal inkjet printhead increases with increasing local temperature near the ejector nozzles. The amount of ink deposited on the page and so the printed image density depends on the droplet mass. Thus, printhead temperature nonuniformity results in printed image density variations that can be unaccept...

The linear stability of parallel shear flows of incompressible viscous fluids is classically described by the Orr–Sommerfeld equation in the disturbance streamfunction. This fourth-order equation is obtained by eliminating the pressure from the linearized Navier–Stokes equation. Here we consider retaining the primitive velocity-pressure formulation...

Air-flow oscillations in inkjet print-zones give rise to undesirable print defects due to misplacement of low Stokes number satellite droplets. The problem is studied numerically using a novel dispersed-phase continuum method, that, due to the separation of length scales, permits the force exerted by the multitude of high Stokes number main droplet...

A numerical model was employed to investigate the vortex instability in a two-dimensional inkjet print-zone. The simulation models the entrainment effect of the droplets on the airflow via a dispersed-phase continuum method that, due to the separation of length scales, treats the force exerted by the main droplets as a continuum smooth field. The t...

Thermal inkjet actuators operate by passing a current through an electrical resistor, the heater, in contact with a working liquid such as ink. This heats the liquid rapidly (dT /dt ≈ 10 8 − 10 9 K/s), and once the liquid temperature exceeds the superheat limit, it flash boils. The resultant high temperature (≈ 300 • C) and pressure (≈ 10 MPa) bubb...

The quality of inkjet printing is impacted by the airflow dynamics in the near vicinity of the printing zone. In particular, at elevated pen-to-paper spacings, (PPS) which is crucial for certain applications, the flow within the print gap of an inkjet printer becomes unstable. This can induce a temporal and spatial periodic misplacement of satellit...

Air-flow oscillations in inkjet print-zones give rise to undesirable print defects due to misplacement of low Stokes number satellite droplets. The problem is studied numerically using a novel body-force approach which averages the entrainment, due to the multitude of high Stokes number main droplets, over the print-zone volume. The instability see...

Partial differential equations (PDEs)—such as the Navier–Stokes equations in fluid mechanics, the Maxwell equations in electromagnetism, and the Schrödinger equation in quantum mechanics—are the basic building blocks of modern physics and engineering. The finite element method (FEM) is a flexible computational technique for the discretization and s...

Flow oscillations in inkjet print-zones give rise to undesirable print defects due to misplacement of low Stokes number satellite droplets. The problem is studied numerically using a novel body-force approach which averages the entrainment, due to the multitude of high Stokes number main droplets, over the printzone volume. The instability seems to...

The stability properties of a natural convection boundary layer adjacent to an isothermally heated vertical wall, with Prandtl number 0.71, are numerically investigated in the configuration of a temporally evolving parallel flow. The instantaneous linear stability of the flow is first investigated by solving the eigenvalue problem with a quasi-stea...

Conjugate heat transfer (CHT) in thermal inkjet printheads is investigated using numerical simulations. The simulation code is compared against analytical solutions to relatively simple problems: fully developed forced convection in a planar channel, and conduction against convection in plug flow. Comparisons are also made with earlier numerical st...

Numerical simulation permits exploration of a system design space prior to manufacture. Here, we present and discuss the results of numerical simulations of the droplet ejection process in thermal inkjet printers. We compare the simulation results with those from prior experimental and numerical investigations, and then consider the performance of...

Numerical simulations of droplet impact have been performed using a two-phase VOF-based solver. Comparison with experimental data for droplet shape, maximum diameter and minimum height is reasonable. Simulation results for maximum droplet diameter, for a range of conditions pertinent to inkjet printing, have been compared with several empirical cor...

A widespread model of thermal ink-jet actuation prescribes the pressure of the expanding bubble as a rapidly decreasing function of time, this being taken as a finite version of a Dirac impulse with only the integral with respect to time mattering. Here it is shown that the impulse is actually easier to treat numerically than the bounded approximat...

The viscosity and surface tension of aqueous mixtures of interest to the inkjet printer designer are estimated using the corresponding states principle and the universal quasi-chemical functional-group activity coefficient method. The former method performs best for estimating viscosity, whilst the latter method works best for surface tension, for...

The problem of tiger striping is described: oscillations of the print-jet cause re-deposition of small aerosols far from their intended location on the print-media. Experimental and numerical studies of the problem have shown the main factors and possible mitigation strategies for this print quality artefact.

Direct numerical simulation is employed to investigate the two-dimensional boundary layer instability of a natural convection flow on a uniformly heated vertical plate submerged in a homogeneous quiescent environment. A Boussinesq fluid with Prandtl numbers of Pr = 0.733 (air) and 6.7 (water), in the local Rayleigh number range 0 ⩽ Rax ⩽ 2.4 × 1010...

Starting from a basic knowledge of mathematics and mechanics gained in standard foundation classes, Theory of Lift: Introductory Computational Aerodynamics in MATLAB/Octave takes the reader conceptually through from the fundamental mechanics of lift to the stage of actually being able to make practical calculations and predictions of the coefficien...

We continue our study of the construction of numerical methods for solving two-point boundary value problems using Green functions, building on the successful use of split-Gauss-type quadrature schemes. Here we adapt the method for eigenvalue problems, in particular the Orr–Sommerfeld equation of hydrodynamic stability theory. Use of the Green func...

The behaviour of plane fountains, resulting from the injection of a denser fluid upwards into a large body of a lighter homogeneous fluid, is investigated numerically. The transient behaviour of fountains with a uniform inlet velocity, Reynolds number Re=100, Prandtl number Pr=7, and Froude number 0.25⩽Fr⩽10.0 is studied numerically. In the present...

Experimental evidence for previously unreported fountain behaviour is presented. It has been found that the first unstable mode of a three-dimensional round fountain is a laminar flapping motion that can grow to a circling or multimodal flapping motion. With increasing Froude and Reynolds numbers, fountain behaviour becomes more disorderly, exhibit...

Natural convection flow in rectangular cavities with uniform heat flux side walls and an adiabatic floor and ceiling is investigated. The analytical solution for the evenly heated and cooled infinitely tall cavity , obtained by integrating the energy equation over a certain control volume, is introduced and compared to a full numerical solution for...

Natural convection flow in rectangular cavities with uniform heat flux side walls and an adiabatic floor and ceiling is investigated. The analytical solution for the evenly heated and cooled infinitely tall cavity, obtained by integrating the energy equation over a certain control volume, is introduced and compared to a full numerical solution for...

Natural convection ow in rectangular cavities with uniform heat ux side walls and an adiabatic oor and ceiling is investigated. The analytical solution for the evenly heated and cooled infinitely tall cav-ity, obtained by integrating the energy equation over a certain control volume, is introduced and compared to a full numerical solution for the f...

We continue our study of the construction of numerical methods for solving two-point boundary value problems using Green's func-tions, building on the successful use of product integration to achieve the convergence expected of Gauss-type quadrature schemes. We in-troduce refinements such as the use of cardinal basis functions to elim-inate the nee...

Experimental evidence for previously unreported fountain behaviour is presented. It has been found that the first unstable mode of a wall bounded three dimensional round fountain is a laminar flapping motion that can grow to a circling or multi-modal flapping motion. With increasing Froude and Reynolds numbers, fountain behaviour becomes more disor...

Stability of natural convection flow adjacent to a vertical plate with uniform heat flux is studied using direct numerical simulation (DNS). Stability of the fully developed boundary layer is examined by adding a perturbation to the simulation in the region of the plate origin, with the resulting traveling waves tracked as they travel up the bound...

In this paper we present the behaviour of plane fountains, injected into a homogeneous fluid of lower density, impinging on a ceiling. The transient behaviour of the impinging fountain with Reynolds number 100 ≤ Re ≤ 1000, Prandtl number Pr = 7, and Froude number Fr = 4 and 5 is studied by direct numerical simulation using a fractional-step solutio...

It is shown how to decompose a three-dimensional field periodic in two Cartesian coordinates into five parts, three of which are identically divergence-free and the other two orthogonal to all divergence-free fields. The three divergence-free parts coincide with the mean, poloidal and toroidal fields of Schmitt and Wahl; the present work, therefore...

We continue our study of the adaptation from spherical to doubly periodic slot domains of the poloidal-toroidal representation of vector fields. Building on the successful construction of an orthogonal quinquepartite decomposition of doubly periodic vector fields of arbitrary divergence with integral representations for the projections of known vec...

When the Green’s function for a two-point boundary value problem can be found, the solution for any forcing term reduces to a quadrature. Here we investigate using this as the basis for numerical schemes for boundary value problems and parabolic initial-boundary value problems. Application of Gaussian quadrature is disappointing, only converging sl...

We treat natural convection in a plane vertical rectangular cavity with one vertical wall evenly heated, the other cooled, and an adiabatic floor and ceiling. We predict the temperature difference across the cavity by simple analysis, and present numerical solutions verifying the results. If the cavity is infinitely tall, there is an exact steady o...

A scaling analysis is used to obtain a description of the basic structure of the boundary layer flow adjacent to a vertical semi-infinite evenly heated plate in the presence of a stably stratified ambient fluid. The scaling analysis provides relations between such quantities as the development time and boundary layer thickness and the fluid propert...

Analytical solutions for the velocity and temperature in an evenly heated and cooled slot are compared to numerical solutions for cavities with vertical aspect ratios of 1 to 10 and Rayleigh numbers of 1×10² – 1×10^7. It is shown that for high Rayleigh number or aspect ratios the numerical solution at mid-height in the cavity and the cavity stratif...

The linear stability of natural convection in a fluid between verti-cal hot and cold walls was studied using a collocation method. Seven figure accurate results for monotonic disturbances were obtained by Ruth (1979) using numerical power series, but this method is intrinsi-cally limited and failed for Pr > 10 . In contrast, Chebyshev colloca-tion...

An easily implemented algorithm is described for tracing the margin of a plane region defined by a predicate. Given a point inside and one outside, a sequence of marginal points is produced. The algorithm is a modified specialization of the ‘simplicial decomposition’ method for n equations in n+1 dimensions. The case n=1 has special properties and...

A collocation technique is applied to the equations governing the linear stability of the anabatic layer on an evenly heated vertical wall in a stratified fluid. Marginal stability curves and critical heat fluxes are obtained for Prandtl numbers from 0 to 1000. As in other cases of vertical natural convection, two kinds of instability are observed,...

Natural convection in horizontally heated spherical fluid-filled cavities is considered in the low Grash of number limit. The equations governing the asymptotic expansion are derived for all orders. At each order a Stokes problem must be solved for the momentum correction. The general solution of the Stokes problem in a sphere with arbitrary smooth...

Scalar fields satisfying the stationary advection–diffusion equation with no source or sink terms cannot have strong local extrema. This can be deduced from the elliptical nature of the equation. Here, however, an alternative, original and more physically motivated proof is offered. It highlights the positive role of diffusion in preventing extrema...

The rate of evaporation from the wetted floor of a tube open at the top to a relatively dry environment is investigated analytically, numerically and experimentally for the case of a light vapour. Though buoyancy forces ensure that the heavier external gas is always in motion, it is found that inside the tube both stagnation (diffusion-dominated ev...

The fully developed flow in a vertical cavity or duct subject to horizontal heating is considered. Solutions of the Boussinesq equations are obtained for rectangular and elliptic sections, in terms of Fourier series and polynomials, respectively. Both generalize the familiar odd-symmetric cubic profile of the plane cavity. Uniqueness is demonst...

Closed form expressions for the fully developed velocity, temperature and concentration profiles in a vertical channel are found by solving the equations for a cavity in the limit as the aspect ratio tends to infinity. We consider plane, steady, laminar, Boussinesq flow of an ideal gas-vapour mixture. The vertical walls are held at different consta...

Simple formulae for the overall heat and moisture transport rates due to laminar natural convection in a rectangular cavity are obtained by scale analysis from the governing differential equations and a simplified picture of the flow. The two formulae contain a single unknown proportionality constant, which is determined by a least squares fit to t...

The three-dimensional structure of the thermal boundary layer adjacent to an evenly heated vertical wall with imposed stable background density stratification is investigated using direct numerical simulation. The wall is considered to extend infinitely in the vertical and spanwise directions, with the heating imposed as a constant flux boundary co...